Yes, there is an internet cafe onboard, the costs are usually around $.75 per minute, so use it wisely. I've found that most ports have internet cafe's by the piers and much cheaper, used one in St. Thomas a few weeks ago for $5.00 for an hour.
Going to St. Thomas? Try this little secret. At the main store for Little Switzerland there is a nice little coffee shop upstairs. They have 4 or five computers with FREE internet access. You do not really have to buy anything to use it. It is a nice quiet place to get away from the crowd and have coffe or a cold drink however.